The Core as a Heat Engine (Thermodynamics focus)

In the center of the earth is a solid Core of iron the size of Pluto, temperature about 5500K, but the solid core is only about a billion years old against the core itself which is about 4.5 billion years old.

How the internals of earth look like is with analysis of P- and S-Waves that travel through the earth. The S-Waves cannot travel through the liquid outer core and get reflected from the inner core.

The earth can be approximated as a Carnot heat engine, but instead of a piston moving, the actual work done is convection. It needs a hot reservoir and a cold sink. In the earth the hot reservoir is the hot internal core and the cold sink is the cooler rocky mantle.

Secular cooling, the constant loss of energy

The main process is secular cooling, it is constant cooling since it has been formed. The mantle is throttling the cooling and dictates the rate of the cooling, it is solid but over long stretches of time it can flow, hotter rock rises and colder rock sinks with a rate of cm per year.

The heat is transferred out of the core via conduction and convection. Convection does do work, it does move material up against gravity.

The total heat flux moving out of the core is between 6 and 14 terawatts (half/two-thirds of the human current demand), but the main heat coming out at the surface is generated in the crust and mantle via radioactive decay of elements, only about 15-35 % is coming from the core.

The Carnot Heat Engine named Terra

The hot reservoir and the cold sink

At the inner boundary the temperature is 5500 Kelvin (about the temperature of the surface of the sun), this temperature was found be crushing iron alloy between two diamonds and heating it up with a laser then see when it melts. So the melting curve of iron can be mapped.

At the outer boundary the temperature is about 4000 Kelvin. This equates to a 1500 Kelvin drop in a distance of 2200 km.

The Throttle to the System

Now the adiabatic heat gradient comes into play, when from the center a parcel of iron rises up through the liquid outer core, the pressure drops and it expanded. In an adiabatic situation this also means the temperature inside the parcel drops. In the outer core this is about 0.8 Kelvin per km.

This adiabatic heat gradient comes from conduction, this heat gradient of 0.8 Kelvin per km can be sustained with just conduction moving heat out of the inner core. This results to about 4 TW that gets moved out of the core with just conduction. Conduction does not do work, it does not generate buoyancy and move the material up, it does not contribute to the Carnot heat engine.

The Fuel

The inner core is slowly growing. As the core loses heat to the mantle the total temperature drops. Now because the pressure is higher near the center, which lowers the melting point, the freezing is happening at the inner boundary. The growth of the inner core is about 0.3 to 1 mm a year.

When a liquid freezes it releases latent heat, which heats up the liquid iron around.

The Age Paradox

But now if only with this calculated the timing does not add up, if this is the only process going on, then the earth would have been extremely hot in the beginning if we run it backwards with this cooling rate. An explanation could be radioactive elements, especially potassium-40 which is siderophile at high-temperature and high-pressure. This potassium releases heat when it decays, so the cooling rate is different, and it stretches the history up to an age of the core of about 3 billion years.

Where Chemistry comes into play

In the outer core there is still a big amount (10 - 15%) of lighter elements (S, O, Si), this changes the compositional buoyancy. When the liquid iron freezes, it cannot hold these lighter elements, they get rejected into the liquid iron. This makes the iron near the boundary lighter and increases convection, not just based on temperature but also because of the compositional density. The difference in density about the inner boundary is about 500 - 900 kg per m3. A third is the physical difference between solid and liquid, the rest is because of compositional density.

The Flowing inside a rotating Sphere

The Coriolis force is so dominant that the fluid has to obey the Taylor-Proudman theorem. This says that the fluid resists the motion along the axis of rotation, but buoyancy wants to move outwards. This forms Taylor columns, they are parallel to the axis of the earth and are like rotating columns.

The Ceiling

The D'' layer is where the core meets the mantle. It is about 1000 Kelvin cooler than the 4000 Kelvin of the Core, this temperature gradient squeezes into 200 km of space.

The heat does not get out of the D'' layer evenly, some regions are covered with subducted tectonic plates like cold mountains and other regions are really hot and partially molten.

This generates mantle plumes and makes volcanic hotspots in for example Hawaii, Iceland or Yellowstone.

In the more isolated regions the heat has nowhere to go, it retains heat and kills convection.